Motivic cohomology of mixed characteristic schemes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916853358526464 |
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| author | Bouis, Tess |
| author_facet | Bouis, Tess |
| contents | We introduce a theory of motivic cohomology for quasi-compact quasi-separated schemes, which generalises the construction of Elmanto--Morrow in the case of schemes over a field. Our construction is non-$\mathbb{A}^1$-invariant in general, but it uses the classical $\mathbb{A}^1$-invariant motivic cohomology of smooth $\mathbb{Z}$-schemes as an input. The main new input of our construction is a global filtration on topological cyclic homology, whose graded pieces provide an integral refinement of derived de Rham cohomology and Bhatt--Morrow--Scholze's syntomic cohomology. Our theory satisfies various expected properties of motivic cohomology, including relations to étale cohomology and to non-connective algebraic $K$-theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06635 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Motivic cohomology of mixed characteristic schemes Bouis, Tess Algebraic Geometry K-Theory and Homology Number Theory We introduce a theory of motivic cohomology for quasi-compact quasi-separated schemes, which generalises the construction of Elmanto--Morrow in the case of schemes over a field. Our construction is non-$\mathbb{A}^1$-invariant in general, but it uses the classical $\mathbb{A}^1$-invariant motivic cohomology of smooth $\mathbb{Z}$-schemes as an input. The main new input of our construction is a global filtration on topological cyclic homology, whose graded pieces provide an integral refinement of derived de Rham cohomology and Bhatt--Morrow--Scholze's syntomic cohomology. Our theory satisfies various expected properties of motivic cohomology, including relations to étale cohomology and to non-connective algebraic $K$-theory. |
| title | Motivic cohomology of mixed characteristic schemes |
| topic | Algebraic Geometry K-Theory and Homology Number Theory |
| url | https://arxiv.org/abs/2412.06635 |